Combinatorial Theory

Quasi-polar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Non-classical examples of quasi-quadrics have been constructed using a technique called {\em pivoting} in a paper by De Clerck, Hamilton, O'Keefe and Penttila. We introduce a more general notion of pivoting, called switching, and also extend this notion to Hermitian polar spaces. The main result of this paper studies the switching technique in detail by showing that, for \(q\geq 4\), if we modify the points of a hyperplane of a polar space to create a quasi-polar space, the only thing that can be done is pivoting. The cases \(q=2\) and \(q=3\) play a special role for parabolic quadrics and are investigated in detail. Furthermore, we give a construction for quasi-polar spaces obtained from pivoting multiple times. Finally, we focus on the case of parabolic quadrics in even characteristic and determine under which hypotheses the existence of a nucleus (which was included in the definition given in the De Clerck-Hamilton-O'Keefe-Penttila paper) is guaranteed.

Mathematics Subject Classifications: 51E20

Keywords: Projective geometry, quadrics, hyperplanes, quasi-quadrics, intersection numbers